A number is normal in a base if, in its expansion to
that base, all possible digit strings of any given
length are equally frequent. While it is generally
believed that many familiar irrational constants are
normal, normality has only been proven for numbers
expressly invented for the purpose of proving their
In this study we review some of the main results to date.
We then define a new normality criterion, strong
normality, to exclude certain normal but clearly
non-random artificial numbers. We show that strongly
normal numbers are normal but that Champernowne''s
number, the best-known example of a normal number,
fails to be strongly normal.
We also re-frame the question of normality as a
question about the frequency of residue classes of an
increasing sequence of integers modulo some fixed
integer. This leads to the beginning of a detailed
examination of the digits of irrational square roots.